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Mathematical application That have an wide range impact On issue such as space

  2. What is space ? We define space as an area which has neither gravitational force nor the friction. Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four- dimensional continuum known as space time .The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework
  3. Angular Distance Between Two Points on a Sphere
  4. Finding the angular distance between two points on a sphere of radius R , suppose that the two points have a right ascension (RA) of ϕ1 and ϕ2, and a declination (DEC) of and . Alternatively, the angles ϕ and could correspond to geographical longitude and latitude respectively. The angle at the center of the sphere separating the two points is: The arc length on the spherical surface is equal to the radius of the sphere times (in radians). When the two points are extremely close together (or otherwise when is close to unity), different formulas must be used to ensure sufficient precision . 1 2    21cos2cos1cos2sin1sinarccos    cos
  5. Uses of this formula ▪ The main use of this formula is to measure the distance between two points in planet without entering into that planet.
  6. Blackbody Radiation Spectrum, Planck Function
  7. The Blackbody radiation spectrum, or Planck function, can be expressed in a number of ways. There are various choices of units but we use examples with a specific set of units in order to clearly illustrate the physical nature of each function. The notation B denotes the intensity spectrum (per unit solid angle), and the notation u donates the energy density (energy per unit volume). The energy density is related to the intensity simply by multiplying the latter by 4π/c for isotropic radiation, summed over all directions. However, if we are dealing with a general angular distribution of radiation we have to start with the energy density per unit solid angle, in which case we would multiply by 1/c instead of 4π/c. We would then integrate over solid angle. 134   cmcmB c uE  
  8. The above formula we can write in different ways: ▪ In terms of frequency using some of this formula ▪ In terms of wavelength ▪ In terms of energy: 13 2 4   cmcmB hc u   134   HzcmB c u vv  134   ergcmB cE u EE  2     cd dv C v vc hdvdE hvE     
  9. For the blackbody spectrum or Planck function formulas ▪ In terms of frequency ▪ In terms of wavelength ▪ In terms of energy 1112 2 3 1)/exp( 12    steradianHzsergcm kThvc hv Bv 1112 5 2 1)/exp( 12    steradiancmsergcm kThc hc B   1112 32 3 1)/exp( 12    steradianergsergcm kTEhc E BE
  10. In space means in 3D we denote center of mass as where : ▪ ▪ In this way all the coordinates will form. ),,( zyxCM M xm m xm x nn n nn      M xm m xm x nn n nn     
  11. hourmiles r AU M M v / 1 94200 2/12/1               r r c v g2  km M M r cmMr where g g          4822.1 104822.1 8 13        r GM v 2
  12. In layman's terms, it is defined as "the point of no return", i.e., the point at which the gravitational pull becomes so great as to make escape impossible, even for light.
  13. An event horizon is most commonly associated with black holes. Light emitted from inside the event horizon can never reach the outside observer. For a non-spinning (stationary) black hole , the event horizon radius, , is the location (radius) from the center of the black hole for which light originating at radii less than the horizon radius cannot escape to infinity because it would be subject to an infinite red shift. The event horizon radius in this case is synonymous with the Schwarzschild radius. hR
  14. Where a is the dimensionless angular momentum of the black hole, lying between 0 and 1, and is the gravitational radius.  2 11 a R R g h  gR
  15. A radius definedfor a bodyof a givenmass and proportional to that mass, such that if the bodyis smallerthan that radius, the force of gravityis strongenoughto prevent matter andenergyto escape fromwithin that radius.
  16. where G is the gravitational constant, M is the black-hole mass, and c is the speed of light. 2 c GM rg  cmMrg 8 3 104822.1 
  17. In the simplest case, if the radiation source is isotropic and has a total luminosity of L (in say, erg s−1), and the flux measured at a distance r from the source is F (in say, erg cm−2 s−1), then the surface area of the sphere times the flux must be equal to the total luminosity (power) at any distance (all r) from the source: 4πr2F = L Then F = L4πr2 and the latter is the inverse square law. The same basic relation applies if instead of L we have the total number of photons s−1 emitted isotropic ally in all directions, say N0, and instead of F we have the number of photons s−1intercepted per unit area at some distance r from the source, say n(r). That would give the following formula (using concrete units to make it even more clear):      2 12 4 / r sphotonsN sphotonscmrn o  
  18. 3. Kepler’s Third Law The square of the orbital period of a planet in orbit around the sun is proportional to the cube of the semi major axis of the orbit. (The semi major axis is equal to half of the largest symmetry axis of the ellipse
  19. It has been obvious for a long while that the motions of galaxies in groups and clusters, if simply interpreted, imply the existence of a substantial amount of mass in addition to that traditionally associated with individual galaxies
  20. FORMULASFOR LIGHT CROSSING TIMES 1. where r = is the distance. 2. t∼500(r1AU) seconds , Here r = is in astronomical units (AU) 3. Light-crossing time per gravitational radius: c r t  ondsM c r t g sec500~ 8      
  21. Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rocket and other spacecraft
  22. hourmiles r AU M M v skm r AU M M v r GM v / 1 66600~ / 1 30~ 2/12/1 2/12/1                                  r r c v g  2/1 2        a r v v f circular elliptical elliptical
  23. In general, two masses, m and Mwill orbit around the center of mass of the systemand the systemcan be replacedby the motionof the reduced mass, μ≡mM/(m+M). It is important to understandthat the following equationsare valid for elliptical orbits(i.e., not just circular), and for arbitrary masses (i.e., not just for the case for one massmuch lessthanthe other). Theonly restrictionis that the motionis purely dueto gravity and that the motionin nonrelativistic .   2/13 2         mMG a T 
  24.      2/13         mMsesinsolarmas a yearT AU a AU is the semi major axis in units of AU.
  25. 1. The effect whereby the position or direction of an object appears to differ when viewed from different positions, e.g. through the viewfinder and the lens of a camera. 2. The angular amount of parallax in a particular case, especially that of a star viewed from different points in the earth's orbit.
  26. pc ondsarcparallax d d x )sec( 1 sin~~~tan  
  27. The Schwarzschild radius of a black hole is the location of the event horizon from the center of a non-spinning (stationary) black hole. In other words, light originating at radii less than the Schwarzschild radius cannot escape to infinity because it would be subject to an infinite red shift.
  28. AUM M M R cmMR km M M R km M M R c GM R s s s s s 88 8 13 2 2 10 2~ 109644.2 3~ 9644.2 2                        